The concept is discussed in "The Death of the State in Marx and Engels" by Hal Draper (1970), summarized here, which would have been hot off the presses at the time that Newton gave his speech, and probably influenced the discussion in left leaning political circles, whether or not he read it himself. The summary explains that:
Engels’ “On Authority” already mentioned in passing that, as the state
disappeared, “public functions will lose their political character and
be transformed into the simple administrative functions of watching
over the true interests of society”. The same idea appeared just as
incidentally in another work of that year (1872):
Anarchy: that is the great war-horse of their master Bakunin, who has
taken from the socialist systems only the labels. All socialists
understand this by Anarchy: once the aim of the proletarian movement,
the abolition of classes, has been attained, the state power which
serves to keep the great productive majority under the yoke of an
exploiting minority small in numbers, disappears, and the governmental
functions are transformed into simple administrative functions. The
Bakuninist Alliance turns the thing upside-down.
This recalls Saint-Simon’s saying about replacing the state with the
“administration of things”. The state would lose its “political
character”, that is, its repressive and coercive character, especially
in terms of class repression and coercion, but there would still be
“public functions” to be organized. What type of (non-state) social
organism will still be needed to organize these “public functions”?
The question is further highlighted as soon as the state concept is
separated from the wider concept of authority as such-a separation
brought to the fore by the controversy with anarchism. For this means
that the disappearance of the state does not yet imply the
disappearance of all exercise of authority in society. This in turn
implies the continued existence of limited organs of authority of some
kind.
Obviously this invites the kind of semi-fictional speculation about
the future which Marx and Engels resolutely refused to carry on, and
still less were they willing to work out prescriptions for that
future.